Hasse invariants for Hilbert modular varieties

Given a totally real field L of degree g, we construct g Hasse invariants on Hilbert modular varieties in characteristic p and characterize their divisors. We show that these divisors give the type stratification defined by the action of O-L On the alpha (p)-elementary subgroup. Under certain conditions, involving special values of zeta functions. the product of these Hasse invariants is the reduction of an Eisenstein series of weight p - 1.